where \(\chi ^{i} (z,\mathbf {p},t)\)  are expansion coefficients and \(\textbf {e}_i(z,\mathbf {p},t)\)  are orthonormal basis vectors.
Since \(G(z,\mathbf {p},t)\)  is a 16-vector, we need a set of 16 unit vectors. Sheng et al. [1]} considered basis vectors that only depended on \(\textbf {p}_{\bot }\)  for the case of a homogeneous electric field. The point of having such basis is that the they will not be acted on by the operator \(D_t\)  and hence one can close the system in a less complicated way. In order to close the system for the homogeneous field case, Sheng et al. used three basis vectors. However, since we consider a space time-dependent electric field, it turns out we need to define one more unit vector. As we will see, we can express \(G(z,\mathbf {p},t)\)  as
\(G(z,\mathbf {p},t)=\sum _{i=1}^{4} \chi ^i(z,\mathbf {p},t)\textbf {e}_i(p_{\bot })\) 
